I mentioned earlier that it is possible to create tile sets that simulate cellular automata. In these tile sets, a solution may be found without backtracking, by selecting pieces in an order corrseponding the the automata's time direction. Such tile sets might be described as causal. Furthermore these tile sets are deterministic, in that the anti-causal neighbours of a location fully determine which tile must be placed in it.

That causality is an emergent property rather than an assumption (as per cellular automata, and simulation of dynamic systems in general) allows us to think about cases that are ``almost'' deterministic-causal. There are two obvious ways to break strict deterministic causality:

We might add several tiles that will match a given previous configuration. The result would be a random cellular automaton.

We might choose not to include any tile which matches a certain input configuration. This would forbid any past state that would lead inevitably to this configuration. The result would be a tile set that is mostly causal, but with teleological elements.

Combining these two changes would yield a tile set that was ``almost'' deterministicly causal. This would seem to be an interesting generalization. It would be interesting to know if teleological effects would tend to be obscured by random elements on sufficiently large scale in such a tile set.

The importance of this is not clear. It is entirely possible it has no importance. The most obvious possible application would be in modelling quantum-scale phenomena. Entanglement, for example, is straightforward to implement in a near-causal tile-set. The discrete nature of tile-sets is also appealing at this scale. However it is not clear how to account for the most interesting aspect of quantum-scale effects, namely destructive interference of the wave-function.

hmm...

Tile sets can be also used to simulate quantum computers, however they are a little too powerful! They can be used for NP-Hard problems that a quantum computer would not be able to compute. Some restriction on the nature of the tile set is needed. For example, we might require that it be solvable in polynomial time, or obey a conservation of information law.

hmm...

If each potential Feynman diagram connecting inputs to outputs could be turned around to bring the outputs back to the inputs[1], you might get a path that zig-zags back and forth in time some random number of times before finally connecting to the output. This would account for destructive interference. Requires going down to a network model of spacetime instead of a pre-supposed grid, as several different elements in the construct would have to occupy the same position in spacetime.

The number of zigzags involved would probably follow an exponential distribution.

(Zigzags can of course also form closed loops: zero-point fluctuations.)

[1] The weight given the diagram would be multiplied by -1 by this inversion. The inversion would also change the particles involved to their anti-particles. I will need to check that real Feynman diagrams have these symmetries.

meh... certainly it can allow all sorts of quantum effects, but are there tile-sets that just allow such effects?